5th Grade Mathematics Unpacked Contents - NC

[Pages:24]5th Grade Mathematics Unpacked Contents For the new Standard Course of Study that will be effective in all North Carolina schools in the 2018-19 School Year.

This document is designed to help North Carolina educators teach the 5th Grade Mathematics Standard Course of Study. NCDPI staff are continually updating and improving these tools to better serve teachers and districts.

What is the purpose of this document? The purpose of this document is to increase student achievement by ensuring educators understand the expectations of the new standards. This document may also be used to facilitate discussion among teachers and curriculum staff and to encourage coherence in the sequence, pacing, and units of study for grade-level curricula. This document, along with on-going professional development, is one of many resources used to understand and teach the NC SCOS.

What is in the document? This document includes a detailed clarification of each standard in the grade level along with a sample of questions or directions that may be used during the instructional sequence to determine whether students are meeting the learning objective outlined by the standard. These items are included to support classroom instruction and are not intended to reflect summative assessment items. The examples included may not fully address the scope of the standard. The document also includes a table of contents of the standards organized by domain with hyperlinks to assist in navigating the electronic version of this instructional support tool.

How do I send Feedback? Please send feedback to us at feedback@dpi.state.nc.us and we will use your input to refine our unpacking of the standards. Thank You!

Just want the standards alone? You can find the standards alone at .

North Carolina Course of Study ? 5th Grade Standards

Operations & Algebraic Thinking

Write and interpret numerical expressions. NC.5.OA.2 Analyze patters and relationships. NC.5.OA.3

Standards for Mathematical Practice

Number & Operations in Base Ten

Number & OperationsFractions

Measurement & Data

Understand the place value system. NC.5.NBT.1 NC.5.NBT.3 Perform operations with multi-digit whole numbers. NC.5.NBT.5 NC.5.NBT.6 Perform operations with decimals. NC.5.NBT.7

Use equivalent fractions as a strategy to add and subtract fractions. NC.5.NF.1 Apply and extend previous understandings of multiplication and division to multiply and divide fractions. NC.5.NF.3 NC.5.NF.4 NC.5.NF.7

Convert like measurement units within a given measurement system. NC.5.MD.1 Represent and interpret data. NC.5.MD.2 Understand concepts of volume. NC.5.MD.4 NC.5.MD.5

Geometry

Understand the coordinate plane. NC.5.G.1 Classify quadrilaterals. NC.5.G.3

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Standards for Mathematical Practice

Practice 1. Make sense of problems

and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

Explanation and Example

Mathematically proficient students in grade 5 should solve problems by applying their understanding of operations with whole numbers, decimals, and fractions including mixed numbers. They solve problems related to volume and measurement conversions. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, "What is the most efficient way to solve the problem?", "Does this make sense?", and "Can I solve the problem in a different way?". Mathematically proficient students in grade 5 should recognize that a number represents a specific quantity. They connect quantities to written symbols and create a logical representation of the problem at hand, considering both the appropriate units involved and the meaning of quantities. They extend this understanding from whole numbers to their work with fractions and decimals. Students write simple expressions that record calculations with numbers and represent or round numbers using place value concepts. In fifth grade mathematical proficient students may construct arguments using concrete referents, such as objects, pictures, and drawings. They explain calculations based upon models and properties of operations and rules that generate patterns. They demonstrate and explain the relationship between volume and multiplication. They refine their mathematical communication skills as they participate in mathematical discussions involving questions like "How did you get that?" and "Why is that true?" They explain their thinking to others and respond to others' thinking. Mathematically proficient students in grade 5 experiment with representing problem situations in multiple ways including numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph, creating equations, etc. Students need opportunities to connect the different representations and explain the connections. They should be able to use all of these representations as needed. Fifth graders should evaluate their results in the context of the situation and whether the results make sense. They also evaluate the utility of models to determine which models are most useful and efficient to solve problems. Mathematically proficient fifth graders consider the available tools (including estimation) when solving a mathematical problem and decide when certain tools might be helpful. For instance, they may use unit cubes to fill a rectangular prism and then use a ruler to measure the dimensions. They use graph paper to accurately create graphs and solve problems or make predictions from real world data. Mathematically proficient students in grade 5 continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to expressions, fractions, geometric figures, and coordinate grids. They are careful about specifying units of measure and state the meaning of the symbols they choose. For instance, when figuring out the volume of a rectangular prism they record their answers in cubic units. In fifth grade mathematically proficient students look closely to discover a pattern or structure. For instance, students use properties of operations as strategies to add, subtract, multiply and divide with whole numbers, fractions, and decimals. They examine numerical patterns and relate them to a rule or a graphical representation. Mathematically proficient fifth graders use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and their prior work with operations to understand algorithms to fluently multiply multidigit numbers and perform all operations with decimals to hundredths. Students explore operations with fractions with visual models and begin to formulate generalizations.

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Operations and Algebraic Thinking

Write and interpret numerical expressions.

NC.5.OA.2 Write, explain, and evaluate numerical expressions involving the four operations to solve up to two-step problems. Include expressions involving:

? Parentheses, using the order of operations.

? Commutative, associative and distributive properties.

Clarification

Checking for Understanding

This standard calls for students to verbally describe the relationship between Write an expression for the number of points Eric has at the end of the game.

expressions without actually calculating them. Students will also need to apply Do not evaluate the expression. The expression should keep track of what

their reasoning of the four operations as well as place value while describing happens in each step listed below.

the relationship between numbers. The standard does not include the use of variables, only numbers and signs for operations.

? John is playing a video game. At a certain point in the game, he has 32,700 points. Then, the following events happen, in order:

? He earns 1760 additional points.

? He loses 4890 points.

? The game ends, and his score doubles.

? John's sister Erica plays the same game. When she is finished

playing, her score is given by the expression: 4(31,500 + 2560) ?

8760.

? Describe a sequence of events that might have led to Erica earning

this score.

Adapted from Illustrative Mathematics ()

Below is a picture that represents 7 + 4

? Draw a picture that represents 3 x (7 + 4) ? How many times bigger is the value of 3 x (7 + 4) than 7 + 4? Explain

your reasoning.

Possible responses:

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The value of 3 x (7 + 4) is three times the value of 7 + 4. We can see this in the picture since 3 x (7 + 4) is visually represented as 3 equal rows with 7 + 4 squares in each row.

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Write and interpret numerical expressions.

NC.5.OA.2 Write, explain, and evaluate numerical expressions involving the four operations to solve up to two-step problems. Include expressions involving:

? Parentheses, using the order of operations.

? Commutative, associative and distributive properties.

Clarification

Checking for Understanding

In this type of picture, the stuent shows that the numbers 7 + 4 are represented by the number of objects, and the number of groups represents the multiplier. Adapted from Illustrative Mathematics ()

Describe how the expression 5(10 x 10) relates to 10 x 10.

Possible response: The expression 5(10 x 10) is 5 times larger than the expression 10 x 10 since I know that I that 5(10 x 10) means that I have 5 groups of (10 x 10).

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Analyze patterns and relationships.

NC.5.OA.3 Generate two numerical patterns using two given rules.

? Identify apparent relationships between corresponding terms.

? Form ordered pairs consisting of corresponding terms from the two patterns.

? Graph the ordered pairs on a coordinate plane.

Clarification

Checking for Understanding

This standard extends the work from Fourth Grade, where students generate Describe the pattern:

numerical patterns when they are given one rule. In Fifth Grade, students are Since Terri catches 4 fish each day, and Sam catches 2 fish, the amount of

given two rules and generate the terms in the resulting sequences. Students Terri's fish is always greater. Terri's fish is also always twice as much as Sam's

should identify, record, and graph ordered pairs on a coordinate plane (first fish. Today, both Sam and Terri have no fish. They both go fishing each day.

quadrant only). After graphing the ordered pairs for each rule, students can Sam catches 2 fish each day. Terri catches 4 fish each day. How many fish do

analyze the relationship between the results.

they have after each of the

Days Sam's Total

Terri's Total

five days? Make a graph of the number of fish. Plot the

Number of Fish

Number of Fish

points on a coordinate plane 0

0

0

and make a line graph, and then interpret the graph.

1

2

4

2

4

8

Make a chart (table) to

3

6

12

represent the number of fish

4

8

16

that Sam and Terri catch.

5

10

20

Student: My graph shows that Terri always has more fish than Sam. Terri's fish increases at a higher rate since she catches 4 fish every day. Sam only catches 2 fish every day, so his number of fish increases at a smaller rate than Terri.

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Analyze patterns and relationships.

NC.5.OA.3 Generate two numerical patterns using two given rules.

? Identify apparent relationships between corresponding terms.

? Form ordered pairs consisting of corresponding terms from the two patterns.

? Graph the ordered pairs on a coordinate plane.

Clarification

Checking for Understanding

Cora and Cecilia each use chalk to make their own number patterns on the

sidewalk. They make each of their patterns 10 boxes long and line their

patterns up so they are next to each other. Cora puts 0 in her first box and

decides that she will add 3 every time to get the next number. Cecilia puts 0 in

her first box and decides that she will add 9 every time to get the next number.

a. Complete each girl's sidewalk pattern.

b. How many times greater is Cecilia's number in the 5th box than Cora's number in the 5th box? What about the numbers in the 8th box? The 10th box?

c. What pattern do you notice in your answers for part b? Why do you think that pattern exists?

d. Write your data as ordered pairs and graph the points on a coordinate plane.

e. What pattern do you notice about your graph? Why do you think that pattern exists?

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Number and Operations in Base Ten

Understand the place value system.

NC.5.NBT.1 Explain the patterns in the place value system from one million to the thousandths place.

? Explain that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it

represents in the place to its left.

? Explain patterns in products and quotients when numbers are multiplied by 1,000, 100, 10, 0.1, and 0.01 and/or divided by 10 and 100.

Clarification

Checking for Understanding

In this standard, students extend their understanding of the base-ten system Danny and Delilah were playing a game where they drew digits and placed

and the magnitude of digits in a number to the relationship between adjacent them on a game board. Danny built the number 247. Delilah built the number

places. This standard also extends student understanding of the relationships 724.

of digits in whole numbers to the relationship of decimal fractions. Students should work with the idea that the tens place is ten times as much as the ones place, and the ones place is 1/10 the size of the tens place.

For example: In the number 55.55, each digit is 5, but the value of the digits is different because of the placement. The 5 that the arrow points to is 1/10 of the 5 to the left and 10 times the 5 to the right. The 5 in the

? How much bigger is the 2 in Danny's number than the 2 in Delilah's number?

? How much smaller is the 4 in Delilah's number than the 4 in Danny's number?

? Write a sentence explaining how the size of the 7 in Danny's number compares to the size of the 7 in Delilah's number.

ones place is 1/10 of 50 and 10 times five tenths.

In class Veronica told her teacher that when you multiply a number by 10, you

just always add 0 to the end of the number. Think about her statement

(conjecture), then answer the following questions.

? When does Veronica's statement (conjecture) work?

? When doesn't Veronica's statement (conjecture) work?

? Is the opposite true? When you divide a number by 10, can you just remove a 0 from the end of the number? When does that work? When

doesn't that work?

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